2026/07/22 by Eyvindur Ari Palsson, Georgios Psaromiligkos
Mathematics · #math.CA
We study pinned nonempty-interior problems for scalar two-point configurations and for volumes of simplices. For E⊂ℝd, d≥ 2, compact and a smooth scalar configuration map Φ(x,y), whose corresponding localized generalized Radon transforms are nondegenerate Fourier integral operators of smoothing order (d-1)/2, we first note how a calculation due to Greenleaf, Iosevich and Taylor can be used to obtain positive Lebesgue measure of ΔΦy(E)=\Φ(x,y):x∈ E\ for almost every pin y when dim\mathcal H(E)>(d+1)/2. Our first main result is to prove that the corresponding one-frequency-loss estimate for differentiation in the level parameter yields a continuous pinned density, and hence nonempty interior, for almost every pin when d≥3 and dim\mathcal H(E)>(d+2)/2. Concrete applications include generalized norm distances, regular variable-coefficient and Riemannian distances, and dot products or nondegenerate bilinear forms on regular patches. Our principal geometric application concerns volumes of simplices. We prove a cylinder-averaging estimate for triangle areas in ℝd and obtain positive measure for doubly pinned area sets at a dimensional threshold (d+1)/2 and nonempty interior at (d+2)/2. A projection theorem then reduces higher simplex-volume problems to triangle areas. In particular, for 3≤ k ≤ d, if dim\mathcal H(E)>(d+k-1)/2, then for every prescribed base point x0 and every prescribed second vertex y∈ E∖\x0\, the set of k-dimensional volumes generated by x0,y and k-1 further points of E has nonempty interior. Thus the result is doubly strongly pinned in its first two vertices.